Question

Graph a parabola whose x-intercepts are at x=-3 and x=5 and whose minimum value is y=-4

Answer 1

(See explanation for further details)

Explanation:

The standard equation of the parabola is:

`y + 4 = C \cdot (x-k)^(2)`

The formula is now expanded into a the form of a second-order polynomial:

`y + 4 = C\cdot x^(2) -2\cdot C\cdot k \cdot x +C\cdot k^(2)`

`y = C\cdot x^(2) - (2\cdot C \cdot k) \cdot x + (C\cdot k^(2)-4)`

The general equation of the second-order polynomial is:

`x = \frac{2\cdot C \cdot k \pm \sqrt{4\cdot C^(2)\cdot k^(2)-4\cdot C\cdot (C\cdot k^(2)-4)}}{2\cdot C}`

`x = k \pm \frac{\sqrt{C^(2)\cdot k^(2)-C^(2)\cdot k^(2)+4\cdot C}}{C}`

`x = k \pm 2\cdot (√(C))/(C)`

`x = k \pm (2)/(√(C))`

The equations to be solved are presented herein:

`-3 = k -(2)/(√(C))`

`5 = k + (2)/(√(C))`

Now, the solution of the system is:

`-3 +(2)/(√(C)) = 5 -(2)/(√(C))`

`(4)/(√(C)) = 8`

`√(C) = (1)/(2)`

`C = (1)/(4)`

`k = 5 - \frac{2}{\sqrt{(1)/(4) }}`

`k = 1`

The equation of the parabola is:

`y = (1)/(4)\cdot (x-1)^(2) -4`

Lastly, the graphic of the function is included as attachment.